1 BA 275 Quantitative Business Methods Summarizing Quantitative Data The Empirical Rule Experiencing Random Behavior Binomial Experiment Binomial Probability Distribution Agenda Attention: Project 1 is due on Wednesday, 1/25/06.

2 Quiz #2

3 Review Example Upper invisible line: 17.4Lower invisible line: 11.8

4 The Empirical Rule

5 Example A set of data whose histogram is bell shaped yields a sample mean and standard deviation of 50 and 4, respectively. Approximately what proportion of observations Are between 46 and 54? Are between 42 and 58? Are between 38 and 62? Are less than 46? Are less than 58?

6 Example: The Empirical Rule A manufacturer of automobile batteries claims that the average length of life for its grade A battery is 60 months with a standard deviation of 10 months. 3 cars in your family used this brand of batteries and none of them lasted for more than 30 months. What do you think about the manufacturer’s claim?

7 Example A manufacturer produces wires with a mean diameter of 1000 microns, and a standard deviation of 1 micron. Is a wire of diameter 1050 microns: Fairly likely? Pretty unlikely? Wildly implausible?

8 Example Suppose that the average hourly earnings of production workers over the past three years were reported to be $12.27, $12.85, and $13.39 with the standard deviations $0.15, $0.18, and $0.23, respectively. The average hourly earnings of the production workers in your company also continued to rise over the past three years from $12.72 in 2002, $13.35 in 2003, to $13.95 in Assuming the distribution of the hourly earnings for all production workers is mound-shaped, demonstrate quantitatively why the earnings in your company become less and less competitive.

9 Review Example Year Industry average Industry std. % increase Company average % increase Z score % % % %2.43

10 What statistical lesson can we learn? “Should we scare the opposition by announcing our mean height or lull them by announcing our median height?”

11 Coin-Tossing Example n = 10 p = 0.5 X = no. of tails in n trials Questions P(X = 5) P(X = 10) P(X < 4) P(X > 8) Business Applications?

12 Characteristics of a Binomial Experiment The experiment consists of n identical trials. There are only two possible outcomes on each trial. We will denote one outcome by S (for Success) and the other by F (for Failure). The probability of S remains the same from trial to trial. This probability is denoted by p, and the probability of F is denoted by 1 – p. The trials are independent. The binomial random variable X is the number of S’s in n trials.

13 Is it a Binomial Experiment? Flip a fair coin 50 times. Toss a fair die 20 times. A multiple-choice quiz has 15 questions. Each question has five possible answers, of which only one is correct. Time is running out and you quickly guess all 15 questions without reading them. You read every question carefully and answer them to the best of your knowledge. Among these questions, three of them are main questions each with 4 subsequent questions. If you don't know the answer to one question, you won't be able to answer the subsequent questions correctly.

14 Probability Distribution All possible outcomes of an experiment The likelihood of observing a particular outcome

15 Binomial Formula and Distribution

16 Example A sign at a gas station claims that one out of four cars needs to have oil added. If the claim is true, what is the probability of the following events? One out of the next four cars needs oil. Two out of the next eight cars need oil.